h-index

equivalent to: distance to maximum degree

Definition: Maximum $h$ for which a graph contains $h$ vertices of degree at least $h$.

Relations

Other		Relation from	Relation to
acyclic chromatic number	red■	exclusion	upper bound
arboricity	red■	exclusion	upper bound
average degree	red■	exclusion	upper bound
average distance	blue■	exclusion	exclusion
bandwidth	green■	upper bound	exclusion
bipartite	blue■	unbounded	exclusion
bipartite number	magenta■	exclusion	unknown to HOPS
bisection bandwidth	blue■	exclusion	exclusion
block	blue■	unbounded	exclusion
book thickness	magenta■	exclusion	unknown to HOPS
boolean width	blue■	exclusion	exclusion
bounded components	green■	upper bound	exclusion
boxicity	red■	exclusion	upper bound
branch width	blue■	exclusion	exclusion
c-closure	blue■	exclusion	exclusion
carving-width	green■	upper bound	exclusion
chordal	blue■	unbounded	exclusion
chordality	red■	exclusion	upper bound
chromatic number	red■	exclusion	upper bound
clique cover number	blue■	exclusion	exclusion
clique-tree-width	blue■	exclusion	exclusion
clique-width	blue■	exclusion	exclusion
cluster	blue■	unbounded	exclusion
co-cluster	blue■	unbounded	exclusion
cograph	blue■	unbounded	exclusion
complete	blue■	unbounded	exclusion
connected	blue■	unbounded	exclusion
contraction complexity	green■	upper bound	exclusion
cutwidth	green■	upper bound	exclusion
cycle	green■	upper bound	exclusion
cycles	green■	upper bound	exclusion
d-path-free	blue■	exclusion	exclusion
degeneracy	red■	exclusion	upper bound
degree treewidth	green■	upper bound	exclusion
diameter	blue■	exclusion	exclusion
diameter+max degree	green■	upper bound	exclusion
disconnected	gray■	unknown to HOPS	unknown to HOPS
disjoint cycles	blue■	unbounded	exclusion
distance to bipartite	blue■	exclusion	exclusion
distance to block	blue■	exclusion	exclusion
distance to bounded components	green■	upper bound	exclusion
distance to chordal	blue■	exclusion	exclusion
distance to cluster	blue■	exclusion	exclusion
distance to co-cluster	blue■	exclusion	exclusion
distance to cograph	blue■	exclusion	exclusion
distance to complete	blue■	exclusion	exclusion
distance to disconnected	red■	exclusion	upper bound
distance to edgeless	green■	upper bound	exclusion
distance to forest	blue■	exclusion	exclusion
distance to interval	blue■	exclusion	exclusion
distance to linear forest	green■	upper bound	exclusion
distance to maximum degree	yellow■	upper bound	upper bound
distance to outerplanar	blue■	exclusion	exclusion
distance to perfect	blue■	exclusion	exclusion
distance to planar	blue■	exclusion	exclusion
distance to stars	blue■	exclusion	exclusion
domatic number	red■	exclusion	upper bound
domination number	blue■	exclusion	exclusion
edge clique cover number	blue■	exclusion	exclusion
edge connectivity	red■	exclusion	upper bound
edgeless	green■	upper bound	exclusion
feedback edge set	blue■	exclusion	exclusion
feedback vertex set	blue■	exclusion	exclusion
forest	blue■	unbounded	exclusion
genus	blue■	exclusion	exclusion
girth	blue■	exclusion	exclusion
grid	green■	upper bound	exclusion
h-index	yellow■	equal	equal
inf-flip-width	blue■	exclusion	exclusion
interval	blue■	unbounded	exclusion
iterated type partitions	blue■	exclusion	exclusion
linear clique-width	blue■	exclusion	exclusion
linear forest	green■	upper bound	exclusion
linear NLC-width	blue■	exclusion	exclusion
linear rank-width	blue■	exclusion	exclusion
maximum clique	red■	exclusion	upper bound
maximum degree	green■	upper bound	exclusion
maximum independent set	blue■	exclusion	exclusion
maximum induced matching	blue■	exclusion	exclusion
maximum leaf number	green■	upper bound	exclusion
maximum matching	green■	upper bound	exclusion
maximum matching on bipartite graphs	green■	upper bound	exclusion
mim-width	magenta■	exclusion	unknown to HOPS
minimum degree	red■	exclusion	upper bound
mm-width	blue■	exclusion	exclusion
modular-width	blue■	exclusion	exclusion
module-width	blue■	exclusion	exclusion
neighborhood diversity	blue■	exclusion	exclusion
NLC-width	blue■	exclusion	exclusion
NLCT-width	blue■	exclusion	exclusion
odd cycle transversal	blue■	exclusion	exclusion
outerplanar	blue■	unbounded	exclusion
path	green■	upper bound	exclusion
pathwidth	blue■	exclusion	exclusion
pathwidth+maxdegree	green■	upper bound	exclusion
perfect	blue■	unbounded	exclusion
planar	blue■	unbounded	exclusion
radius-r flip-width	magenta■	exclusion	unknown to HOPS
rank-width	blue■	exclusion	exclusion
shrub-depth	blue■	exclusion	exclusion
sim-width	magenta■	exclusion	unknown to HOPS
size	green■	upper bound	exclusion
star	green■	upper bound	exclusion
stars	blue■	unbounded	exclusion
topological bandwidth	cyan■	unknown to HOPS	exclusion
tree	blue■	unbounded	exclusion
tree-independence number	magenta■	exclusion	unknown to HOPS
treedepth	blue■	exclusion	exclusion
treelength	magenta■	exclusion	unknown to HOPS
treewidth	blue■	exclusion	exclusion
twin-cover number	blue■	exclusion	exclusion
twin-width	blue■	exclusion	exclusion
vertex connectivity	red■	exclusion	upper bound
vertex cover	green■	upper bound	exclusion
vertex integrity	green■	upper bound	exclusion

Results

2019 The Graph Parameter Hierarchy by Sorge
- page 8 : h-index upper bounds acyclic chromatic number by a polynomial function – Lemma 4.7. The acyclic chromatic number $\chi_a$ is upper bounded by the $h$-index $h$. We have $\chi_a \le h(h+1)/2$.
assumed
- h-index is equivalent to h-index – assumed
unknown source
- maximum degree upper bounds h-index by a linear function – As h-index seeks $k$ vertices of degree $k$ it is trivially upper bound by maximum degree.
- h-index upper bounds distance to maximum degree by a linear function – Remove the $h$ vertices of degree at least $h$ to get a graph that has maximum degree $h$.
- distance to maximum degree upper bounds h-index by a linear function – Removal of $k$ vertices yielding a graph with maximum degree $c$ means that there were $k$ vertices of arbitrary degree and the remaining vertices had degree at most $k+c$. Hence, $h$-index is no more than $k+c$.
- graph class stars has unbounded h-index
- star upper bounds h-index by a constant – trivially
- graph class tree has unbounded h-index – trivially
Comparing Graph Parameters by Schröder
- page 12 : distance to linear forest upper bounds h-index by a linear function – Proposition 3.2
- page 25 : bounded treedepth does not imply bounded h-index – Proposition 3.22
- page 25 : bounded feedback edge set does not imply bounded h-index – Proposition 3.22